Sets of Lengths

DOI10.4169/amer.math.monthly.123.10.960zbMath1391.13004arXiv1509.07462OpenAlexW2962843469WikidataQ58122629 ScholiaQ58122629MaRDI QIDQ4576571

Alfred Geroldinger

Publication date: 12 July 2018

Published in: The American Mathematical Monthly (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1509.07462

Mathematics Subject Classification ID

Class field theory (11R37) Fibonacci and Lucas numbers and polynomials and generalizations (11B39) Divisibility and factorizations in commutative rings (13A05) Arithmetic theory of semigroups (20M13)

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Uses Software

• numericalsgps

Cites Work

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• Multiplicative ideal theory and factorization theory. Commutative and non-commutative perspectives. Selected papers based on the presentations at the meeting `Arithmetic and ideal theory of rings and semigroups', Graz, Austria, September 22--26, 2014
• A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
• A quasi-local half-factorial domain with an atomic non-half-factorial integral closure
• Factoring ideals in integral domains
• Factorization theory: from commutative to noncommutative settings.
• Factorization problems in semigroups
• Arithmetic of seminormal weakly Krull monoids and domains
• Factorization in Dedekind domains with finite class group
• Half-factorial subrings of factorial domains
• Combinatorial number theory and additive group theory. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse)
• On the arithmetic of tame monoids with applications to Krull monoids and Mori domains.
• A realization theorem for sets of lengths
• On products of \(k\) atoms.
• On non-unique factorizations into irreducible elements
• Sets of minimal distances and characterizations of class groups of Krull monoids
• A realization theorem for sets of lengths in numerical monoids
• Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids
• Structural additive theory. Based on courses given at Karl-Franzens-Universität Graz, Austria, 2008--2012
• Non-commutative Krull monoids: a divisor theoretic approach and their arithmetic.
• A characterization of class groups via sets of lengths. II.
• Sets of lengths in maximal orders in central simple algebras.
• Factorizations of upper triangular matrices
• Products of two atoms in Krull monoids and arithmetical characterizations of class groups.
• Integer-Valued Polynomials
• From the Fundamental Theorem of Algebra to Kempe's Universality Theorem
• The set of minimal distances in Krull monoids
• Relative Polynomial Closure and Monadically Krull Monoids of Integer-Valued Polynomials
• An Overview of the Computational Aspects of Nonunique Factorization Invariants
• Arithmetic of Mori Domains and Monoids: The Global Case
• Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group
• Factorizations of Elements in Noncommutative Rings: A Survey
• A Tale of Two Monoids: A Friendly Introduction to Nonunique Factorizations
• Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory
• The set of distances in Krull monoids
• Factorization in Krull monoids with infinite class group
• What You Should Know About Integer-Valued Polynomials
• Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids
• Factorization Theory and Decompositions of Modules
• ON v-MAROT MORI RINGS AND C-RINGS
• Matrix Number Theory: An Example of Nonunique Factorization
• ON THE DIVISOR-CLASS GROUP OF MONADIC SUBMONOIDS OF RINGS OF INTEGER-VALUED POLYNOMIALS
• A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems
• On unique factorization domains

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